Q. 35

The pth</sup

Answer :

Let A be the first term and D the common difference of A.P.


Tp = a = A + (p − 1) D = (A − D) + pD …..(1) 1/2


Tq = b = A + (q – 1) D = (A − D) + qD …..(2) 1/2


Tr = c = A + (r − 1) D = (A − D) + rD …..(3) 1/2


Here we have got two unknowns A and D which are to be eliminated.


We multiply (1), (2) and (3) by q − r, r − p and p − q respectively and add:


a (q – r) = (A – D )(q – r) + D p(q – r) 1/2


b(r – p) = (A – D) (r – p) + Dq (r – p) 1/2


c(p – q) = (A – D) (p – q) + Dr (p – q) 1/2


a(q − r) + b(r − p) + c(p − q) 1


= (A − D)[q – r + r – p + p − q] + D[p(q − r) + q(r − p) + r(p − q)]


= (A – D)(0) + D[pq – pr + qr – pq + rp – rq ) 1


= 0


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